Simplify 1/(x^2-64) - (x-5)/(x^2+5x-24) A Step-by-Step Guide

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In this comprehensive guide, we will delve into the process of simplifying the algebraic expression $\frac{1}{x2-64}-\frac{x-5}{x2+5 x-24}$. This type of problem is a staple in algebra, requiring a solid understanding of factoring, finding common denominators, and simplifying rational expressions. Whether you're a student looking to master algebraic manipulation or simply someone brushing up on their math skills, this guide will provide a step-by-step approach to tackle this problem effectively.

Understanding the Building Blocks

Before diving into the simplification process, it's crucial to understand the fundamental concepts involved. The expression we're dealing with involves rational expressions, which are essentially fractions with polynomials in the numerator and denominator. To simplify these expressions, we'll need to employ techniques such as factoring and finding common denominators. Factoring allows us to break down polynomials into simpler expressions, making it easier to identify common factors and simplify fractions. Meanwhile, finding a common denominator is essential for adding or subtracting fractions, just like with regular numerical fractions. The key is to identify the least common multiple (LCM) of the denominators, which will serve as our common denominator. With a strong grasp of these building blocks, we can confidently approach the simplification process.

Step 1: Factoring the Denominators

The first crucial step in simplifying the expression $ rac{1}{x2-64}-\frac{x-5}{x2+5 x-24}$ is to factor the denominators. Factoring allows us to identify common factors, which is essential for finding a common denominator and simplifying the expression. Let's start by factoring the first denominator, x² - 64. This is a difference of squares, which can be factored into the form (a - b)(a + b). In this case, a is x and b is 8, so we have:

x2−64=(x−8)(x+8)x^2 - 64 = (x - 8)(x + 8)

Now, let's move on to the second denominator, x² + 5x - 24. To factor this quadratic expression, we need to find two numbers that multiply to -24 and add up to 5. These numbers are 8 and -3. Therefore, we can factor the expression as follows:

x2+5x−24=(x+8)(x−3)x^2 + 5x - 24 = (x + 8)(x - 3)

By factoring the denominators, we've laid the groundwork for finding a common denominator and simplifying the expression. This step is critical because it allows us to rewrite the fractions with a common base, making the subtraction operation possible. Remember, careful factoring is key to avoiding errors in the subsequent steps. With the denominators factored, we are now ready to move on to the next stage of the simplification process: finding the least common denominator.

Step 2: Finding the Least Common Denominator (LCD)

After successfully factoring the denominators in the expression $\frac{1}{x2-64}-\frac{x-5}{x2+5 x-24}$, the next crucial step is to identify the Least Common Denominator (LCD). The LCD is the smallest expression that is divisible by both denominators. This will allow us to combine the two fractions into one. We've already factored the denominators as follows:

  • x² - 64 = (x - 8)(x + 8)
  • x² + 5x - 24 = (x + 8)(x - 3)

To find the LCD, we need to consider all the unique factors present in the denominators. In this case, the unique factors are (x - 8), (x + 8), and (x - 3). The LCD is the product of these unique factors:

LCD=(x−8)(x+8)(x−3)LCD = (x - 8)(x + 8)(x - 3)

By identifying the LCD, we now have a common denominator that we can use to rewrite both fractions. This is a critical step in the simplification process, as it allows us to combine the fractions and proceed with the subtraction. It's important to ensure that we include all unique factors in the LCD to avoid errors in the subsequent steps. Now that we have the LCD, we can move on to the next step: rewriting the fractions with the common denominator.

Step 3: Rewriting Fractions with the Common Denominator

Now that we have determined the Least Common Denominator (LCD) for the expression $\frac{1}{x2-64}-\frac{x-5}{x2+5 x-24}$, which is (x - 8)(x + 8)(x - 3), the next step is to rewrite each fraction with this common denominator. This involves multiplying the numerator and denominator of each fraction by the factors that are missing from their original denominators. Let's start with the first fraction:

1x2−64=1(x−8)(x+8)\frac{1}{x^2 - 64} = \frac{1}{(x - 8)(x + 8)}

To get the LCD, we need to multiply the denominator by (x - 3). To maintain the fraction's value, we must also multiply the numerator by the same factor:

1(x−8)(x+8)∗(x−3)(x−3)=x−3(x−8)(x+8)(x−3)\frac{1}{(x - 8)(x + 8)} * \frac{(x - 3)}{(x - 3)} = \frac{x - 3}{(x - 8)(x + 8)(x - 3)}

Now, let's move on to the second fraction:

x−5x2+5x−24=x−5(x+8)(x−3)\frac{x - 5}{x^2 + 5x - 24} = \frac{x - 5}{(x + 8)(x - 3)}

In this case, we need to multiply the denominator by (x - 8) to get the LCD. Again, we must multiply the numerator by the same factor:

x−5(x+8)(x−3)∗(x−8)(x−8)=(x−5)(x−8)(x−8)(x+8)(x−3)\frac{x - 5}{(x + 8)(x - 3)} * \frac{(x - 8)}{(x - 8)} = \frac{(x - 5)(x - 8)}{(x - 8)(x + 8)(x - 3)}

By rewriting both fractions with the common denominator, we've set the stage for the next step: subtracting the fractions. This process ensures that we are working with equivalent fractions, making the subtraction operation valid. It's crucial to multiply both the numerator and denominator by the same factors to maintain the value of the fraction. With the fractions now sharing a common denominator, we can proceed with the subtraction in the next step.

Step 4: Subtracting the Fractions

With both fractions now rewritten with the common denominator (x - 8)(x + 8)(x - 3), we can proceed to subtract the fractions in the expression $\frac{1}{x2-64}-\frac{x-5}{x2+5 x-24}$. This involves subtracting the numerators while keeping the common denominator. We have:

x−3(x−8)(x+8)(x−3)−(x−5)(x−8)(x−8)(x+8)(x−3)\frac{x - 3}{(x - 8)(x + 8)(x - 3)} - \frac{(x - 5)(x - 8)}{(x - 8)(x + 8)(x - 3)}

Now, subtract the numerators:

(x−3)−(x−5)(x−8)(x−8)(x+8)(x−3)\frac{(x - 3) - (x - 5)(x - 8)}{(x - 8)(x + 8)(x - 3)}

Next, we need to expand the product in the numerator:

(x−3)−(x2−8x−5x+40)(x−8)(x+8)(x−3)\frac{(x - 3) - (x^2 - 8x - 5x + 40)}{(x - 8)(x + 8)(x - 3)}

Simplify the expression inside the parentheses:

(x−3)−(x2−13x+40)(x−8)(x+8)(x−3)\frac{(x - 3) - (x^2 - 13x + 40)}{(x - 8)(x + 8)(x - 3)}

Now, distribute the negative sign:

x−3−x2+13x−40(x−8)(x+8)(x−3)\frac{x - 3 - x^2 + 13x - 40}{(x - 8)(x + 8)(x - 3)}

Combine like terms in the numerator:

−x2+14x−43(x−8)(x+8)(x−3)\frac{-x^2 + 14x - 43}{(x - 8)(x + 8)(x - 3)}

By subtracting the fractions, we've combined the two rational expressions into a single fraction. This step requires careful attention to the distribution of the negative sign and the combination of like terms. It's important to double-check your work to avoid errors. With the subtraction complete, we can now move on to the final step: simplifying the resulting expression.

Step 5: Simplifying the Resulting Expression

After subtracting the fractions, we arrived at the expression $\frac{-x^2 + 14x - 43}{(x - 8)(x + 8)(x - 3)}$. The final step in simplifying the original expression $\frac{1}{x2-64}-\frac{x-5}{x2+5 x-24}$ is to examine the resulting expression to see if it can be further simplified. This often involves looking for common factors between the numerator and the denominator. In this case, the numerator is a quadratic expression, and the denominator is a product of linear factors.

Let's first consider the numerator, -x² + 14x - 43. We can try to factor this quadratic, but it's not immediately obvious if it has any simple factors. We can use the discriminant (b² - 4ac) to check if it has real roots. In this case, a = -1, b = 14, and c = -43. The discriminant is:

142−4(−1)(−43)=196−172=2414^2 - 4(-1)(-43) = 196 - 172 = 24

Since the discriminant is positive, the quadratic has real roots, but they are not rational (since 24 is not a perfect square). This means the quadratic does not factor nicely using integers. Next, we look at the denominator (x - 8)(x + 8)(x - 3). We need to check if any of these factors are also factors of the numerator. Since the numerator does not factor easily, it's unlikely that it shares any factors with the denominator.

Therefore, the expression is already in its simplest form:

−x2+14x−43(x−8)(x+8)(x−3)\frac{-x^2 + 14x - 43}{(x - 8)(x + 8)(x - 3)}

This final step is crucial because it ensures that we have presented the expression in its most reduced form. While in this case, the expression couldn't be simplified further, it's important to always check for common factors. By completing this step, we have successfully simplified the original expression. The simplified expression is the final result of our step-by-step process.

Conclusion

In conclusion, we have successfully simplified the expression $\frac{1}{x2-64}-\frac{x-5}{x2+5 x-24}$ by following a step-by-step approach. This process involved factoring the denominators, finding the least common denominator, rewriting the fractions with the common denominator, subtracting the fractions, and simplifying the resulting expression. Each step is crucial in ensuring the accuracy of the final result. Mastering these techniques is essential for anyone studying algebra and beyond. While the final expression, $\frac{-x^2 + 14x - 43}{(x - 8)(x + 8)(x - 3)}$, could not be simplified further in this case, the process demonstrates the importance of checking for common factors. This comprehensive guide provides a solid foundation for simplifying rational expressions and tackling similar problems with confidence. Remember, practice makes perfect, so continue to work through various examples to strengthen your skills in algebraic manipulation.